Firefly Flash Synchronization: Pulse-Coupled Model (2D)

Population & Coupling

Fireflies (N) 200

Coupling / kick strength ε 0.070

Natural-frequency spread 12%

Interaction radius 16%

Live Readout

Order parameter r

0.000

Elapsed time

0.0 s

Time to sync (r≥0.9)

—

Flashes / s

0.0

How it works

This is the Mirollo–Strogatz pulse-coupled model (1990) of firefly synchronization — a different mechanism from continuous Kuramoto phase-coupling used elsewhere on this site. Each firefly i carries a phase φᵢ that climbs linearly toward 1 at its own natural rate ωᵢ. Phase maps to a "charge" state through a concave excitability function:

x = f(φ) = ln(1 + (e^b − 1)·φ) / b     (b = 1.4, concave)

φᵢ → φᵢ + ωᵢ·dt
when φᵢ ≥ 1:  fire, φᵢ → 0
  for each unflashed neighbour j within radius R:
    xⱼ → min(1, f(φⱼ) + ε)
    if xⱼ ≥ 1: neighbour j fires too (cascade)
    else:      φⱼ → f⁻¹(xⱼ)

There is no continuous "nudge toward the average" here — coupling happens only at the instant of a flash, as a discrete jump, and a jump that itself pushes a neighbour over threshold triggers an immediate cascade (the "absorption" Mirollo & Strogatz proved always eventually merges every firefly into one firing group, for identical oscillators). The chart samples the classic order parameter r = |mean(e^{i·2πφ})| to quantify how tightly the population's phases have clustered.

Coupling ε — how much a flash advances a nearby unflashed firefly's charge; stronger ε provably shortens time-to-sync.
Frequency spread — how much individual ωᵢ vary; more heterogeneity makes the population slower and less reliable to lock.
Population & radius — both raise the effective connectivity of the pulse-coupling network, which speeds convergence.

Firefly (glow = charge x) Flash (phase reset) Order parameter r(t)